The classical SIR and SIS epidemiological models are extended by considering the number of adequate contacts per infective in unit time as a function of the total population in such a way that this number grows less rapidly as the total population increases. A diffusion term is added to the SIS model and this leads to a reaction–diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the SIR and SIS models, with the disease–free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, for both models, the endemic equilibrium is locally asymptotically stable and traveling wave solutions are found for the SIS diffusion model. Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous SIR and SIS models are presented. In particular, for the SIS model, a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided. On the other hand the SIS model is extended to a Volterra integral equation, for which the existence of multiple endemic equilibria is proved. This fact is confirmed by numerical simulations. / Dissertation (MSc)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted

As traditional communication technology we use in our day-to-day life reaches its limitations, the international community searches for new methods to communicate information. One such novel approach is the so-called molecular communication system. During the last few decades, molecular communication systems become more and more popular. The main difference between traditional communication and molecular communication systems is that in the latter, information transfer occurs through chemical means, most often between microorganisms. This process already happens all around us naturally, for example, in the human body. Even though the molecular communication topic is attractive to researchers, and a lot of theoretical results are available - one cannot claim the same about the practical use of molecular communication. As for experimental results, a few studies have been done on the macroscale, but investigations at the micro- and nanoscale ranges are still lacking because they are a challenging task. In this work, a self-contained introduction of the underlying theory of molecular communication is provided, which includes knowledge from different areas such as biology, chemistry, communication theory, and applied mathematics. Two numerical methods are implemented for three well-studied partial differential equations of the MC field where advection, diffusion, and the reaction are taken into account. Numerical results for test cases in one and three dimensions are presented and discussed in detail. Conclusions and essential analytical and numerical future directions are then drawn.

Yes / The Poisson's equation is an essential entity of applied mathematics for modelling many phenomena of importance. They include the theory of gravitation, electromagnetism, fluid flows and geometric design. In this regard, finding efficient solution methods for the Poisson's equation is a significant problem that requires addressing. In this paper, we show how it is possible to generate approximate solutions of the Poisson's equation subject to various boundary conditions. We make use of the discrete finite difference operator, which, in many ways, is similar to the standard finite difference method for numerically solving partial differential equations. Our approach is based upon the Laplacian averaging operator which, as we show, can be elegantly applied over many folds in a computationally efficient manner to obtain a close approximation to the solution of the equation at hand. We compare our method by way of examples with the solutions arising from the analytic variants as well as the numerical variants of the Poisson's equation subject to a given set of boundary conditions. Thus, we show that our method, though simple to implement yet computationally very efficient, is powerful enough to generate approximate solutions of the Poisson's equation. / Supported by the European Union’s Horizon 2020 Programme H2020-MSCA-RISE-2017, under the project PDE-GIR with grant number 778035.

Ultra-fast laser heating of nano-films is investigated using 3-D Dual Phase Lag heat transport equation with laser heating at different locations on the metal film. The energy absorption rate, which is used to model femtosecond laser heating, is modified to accommodate for three-dimensional laser heating. A numerical solution based on an explicit finite-difference method is employed to solve the DPL equation. The stability criterion for selecting a time step size is obtained using von Neumann eigenmode analysis, and grid function convergence tests are performed. DPL results are compared with classical diffusion and hyperbolic heat conduction models and significant differences among these three approaches are demonstrated. We also develop an implicit finite-difference scheme of Crank-Nicolson type for solving 1-D and 3-D DPL equations. The proposed numerical technique solves one equation unlike other techniques available in the literature, which split the DPL equation into a system of two equations and then apply discretization. Stability analysis is performed using a von Neumann stability analysis. In 3-D, the discretized equation is solved using delta-form Douglas and Gunn time splitting. The performance of the proposed numerical technique is compared with the numerical techniques available in the literature.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18416号 / 情博第531号 / 新制||情||94(附属図書館) / 31274 / 京都大学大学院情報学研究科複雑系科学専攻 / (主査)教授 磯 祐介, 教授 西村 直志, 教授 木上 淳, 講師 吉川 仁 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

In this thesis we study the optimal stochastic control problem of the drift of a Lévy process. We show that, for a broad class of Lévy processes, the partial integro-differential Hamilton-Jacobi-Bellman equation for the value function admits classical solutions and that control policies exist in feedback form. We then explore the class of Lévy processes that satisfy the requirements of the theorem, and find connections between the uniform integrability requirement and the notions of the score function and Fisher information from information theory. Finally we present three different numerical implementations of the control problem: a traditional dynamic programming approach, and two iterative approaches, one based on a finite difference scheme and the other on the Fourier transform. / text

Šiame darbe pristatomi nauji rezultatai, gauti tiriant diskretųjį Šturmo ir Liuvilio uždavinį su viena klasikine o kita nelokaliąja integraline kraštine sąlyga. Pirmoje dalyje pristatomas diferencialinis Šturmo ir Liuvilio uždavinys su nelokaliąja integraline kraštine sąlyga. Šio uždavinio kompleksinė spektro dalis buvo ištirta bakalauro darbe. Antroje darbo dalyje diferencialinis uždavinys suvedamas į antros eilės baigtinių skirtumų schemą, kai nelokalioji integralinė sąlyga aproksimuojama pagal trapecijų arba Simpsono formulę. Ištirta skirtuminių operatorių su nelokaliosiomis kraštinėmis sąlygomis spektro struktūra, tikrinių reikšmių priklausomybė nuo parametrų γ ir ξ esančių nelokaliosiose sąlygose, reikšmių ir pasirinkto tinklo taškų skaičiaus n. Rezultatai pateikiami charakteristinių funkcijų grafikais ir jų projekcijomis. / In this paper we present a new result of the investigation discrete Sturm--Liuoville problem with one classical and the other nonlocal integral boundary condition. The first part of paper presents differential Sturm Liuoville problem with integral boundary condition. Complex part of spectrum for Sturm Liuoville problem with integral boundary condition was investigated in Bachelor Thesis. The second part of paper present result of investigation second-order finite difference scheme, when the integral conditions condition is approximated by the Trapezoid or Simpson's rules. There are investigated the spectrum of the finite-difference schemes and it dependence on the parameters γ and ξ from nonlocal boundary condition n,where n number of grid points. Simulation results are presented as graphs and projections of characteristic functions.

Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of an effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations as well as the optimal linear complexity of computing time. In Chapter 5, we offer a new implicit finite difference scheme in time for solving the first-order necessary optimality system arising in optimal control of wave equations. With a five-point central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy, which is not restricted by the classical Courant-Friedrichs-Lewy (CFL) stability condition on the spatial and temporal step sizes. Moreover, based on its advantageous developed structure, an efficient preconditioned Krylov subspace method is provided and analyzed for solving the discretized sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of proposed preconditioned iterative solver. Finally, brief summaries and future research perspectives are given in Chapter 6.

Magistro darbe pateikiami nauji rezultatai, gauti tiriant diskretųjį Šturmo ir Liuvilio uždavinį su viena klasikine (arba Noimano) ir antra nelokalia dvitaške kraštine sąlyga. Analitinėje dalyje pateikiama teorija, reikalinga nagrinėjamo uždavinio tyrimui ir trumpai pristatomi rezultatai, gauti tiriant panašų uždavinį. Projektinėje dalyje ištirta diferencialinio uždavinio ir baigtinių skirtumų schemų kompleksinės spektro dalies priklausomybė nuo nelokaliųjų kraštinių sąlygų parametrų $\gamma$ ir $\xi$. Dauguma tyrimo rezultatų pateikiama kompleksinės ir realiosios charakteristinių funkcijų grafikais. / In this Master thesis presented new results which are got investigated the Sturm--Liouville problem with one classical (or Neumann) and another two-point nonlocal boundary condition. In the analytical part are presented the theory, which is necessary to study the problem and presented the results of investigation a similar task. In the design part are investigated the spectrum in complex plane depends on the nonlocal boundary conditions parameters $\gamma$ and $\xi$ in differential problem and in the finite difference schemes. Simulation results are presented as graphs of complex-real characteristic functions.

Darbo tikslas — ištirti baigtiniu skirtumu metodo antrosios eiles hiperbolinio tipo diferencialinei lygciai su nelokaliosiomis integralinemis kraštinemis salygomis stabiluma. Siekiant numatyto tikslo buvo sprendžiami šie uždaviniai: • išnagrinetas antrosios eiles hiperbolines lygties trisluoksnes skirtumines schemos suvedimas i dvisluoksne skirtumine schema; • išanalizuotas skirtuminio operatoriaus perejimo matricos spektras; • gauta pakankamoji skirtumines schemos stabilumo salyga, nusakoma nelokaliuju salygu parametrais; • atlikti skaitiniai eksperimentai, patvirtinantys teorines išvadas. Nurodyta stabilumo salyga yra esmine, sprendžiant hiperbolinio tipo uždavinius su pakankamai didelemis T reikšmemis. Skirtuminio operatoriaus perejimo matricos spektro tyrimo metodika gali buti pritaikyta placios klases diferencialiniu lygciu su nelokaliosiomis salygomis stabilumui tirti. / On the stability of an explicit difference scheme for hyperbolic equation with integral conditions. The aim of the work is stability analysis of solution of finite difference method for hyperbolic equations. Trying to achieve formulated aim these tasks were solved: • a method of transformation of three-layered finite difference scheme into two-layered one was investigated; • a spectrum of transition matrix subject to the properties of second order differential operator Lambda was studied; • stability conditions of hyperbolic type equations with nonlocal conditions subject to boundary parameters were obtained; • numerical experiments, confirming theoretical derivations were made. Derived results could be used to solve one-dimensional tasks with hyperbolic equations in different sciences, to analyse spectrum structure of mathematical models and construct new numerical methods for solving hyperbolic PDEs.